stability analysis of stagnation-point flow past a shrinking sheet in a

Journal of Quality Measurement and Analysis Jurnal Pengukuran Kualiti dan Analisis

STABILITY ANALYSIS OF STAGNATION-POINT FLOW PAST A SHRINKING SHEET IN A NANOFLUID

(Analisis Kestabilan bagi Aliran Titik-Genangan melalui Helaian Mengecut dalam Nanobendalir)

AMIN NOOR1, ROSLINDA NAZAR1 & KHAMISAH JAFAR2

ABSTRACT

In this paper, a numerical and theoretical study has been performed for the stagnation-point boundary layer flow and heat transfer towards a shrinking sheet in a nanofluid. The mathematical nanofluid model in which the effect of the nanoparticle volume fraction is taken into account, is considered. The governing nonlinear partial differential equations are transformed into a system of nonlinear ordinary differential equations using a similarity transformation which is then solved numerically using the function bvp4c in Matlab. Numerical results are obtained for the skin friction coefficient, the local Nusselt number as well as the velocity and temperature profiles for some values of the governing parameters, namely the nanoparticle volume fraction φ , the shrinking parameter l and the Prandtl number Pr. Three different types of nanoparticles are considered, namely Cu, Al2O3 and TiO2. It is found that solutions do not exist for larger shrinking rates and dual (upper and lower branch) solutions exist when l < -1.0. A stability analysis has been performed to determine which branch solutions are stable and physically realisable. It is also found that the upper branch solutions are stable while the lower branch solutions are unstable.

Keywords: heat transfer; nanofluid; shrinking sheet; stability analysis; stagnation-point flow

ABSTRAK

Dalam makalah ini, kajian secara teori dan berangka telah dilakukan bagi aliran lapisan sempadan titik-genangan dan pemindahan haba terhadap helaian mengecut dalam nanobendalir. Model nanobendalir bermatematik dengan mengambil kira kesan pecahan isi padu nanozarah telah dipertimbangkan. Persamaan pembezaan separa menakluk tak linear dijelmakan kepada suatu sistem persamaan pembezaan biasa tak linear menggunakan penjelmaan keserupaan yang kemudiannya diselesaikan secara berangka dengan fungsi bvp4c dalam Matlab. Keputusan berangka diperoleh bagi pekali geseran kulit, nombor Nusselt setempat, serta profil-profil halaju dan suhu bagi beberapa nilai parameter menakluk seperti pecahan isi padu nanozarah φ , parameter mengecut l dan nombor Prandtl Pr. Tiga jenis nanozarah berbeza yang dipertimbangkan adalah Cu, Al2O3 dan TiO2. Didapati bahawa penyelesaian tidak wujud bagi kadar pengecutan yang lebih besar dan penyelesaian dual (cabang atas dan bawah) wujud apabila l < -1.0. Analisis kestabilan dilakukan bagi menentukan penyelesaian cabang yang stabil dan bermakna secara fizikal. Didapati juga bahawa penyelesaian cabang atas adalah stabil manakala penyelesaian cabang bawah tidak stabil.

Kata kunci: pemindahan haba; nanobendalir; helaian mengecut; analisis kestabilan; aliran titik-genangan

1. Introduction

Heat transfer is an important process in physics and engineering, and consequently improvements in heat transfer characteristics will improve the efficiency of many processes. Nanofluid is a new class of heat transfer fluids that comprise of a base fluid and nanoparticles. The use of additive is a technique applied to enhance the heat transfer performance of base fluids. Nanofluids have been shown to increase the thermal conductivity and convective

JQMA 10(2) 2014, 51-63

Amin Noor, Roslinda Nazar & Khamisah Jafar

52

heat transfer performance of the base liquid (Abu-Nada 2008). Thus nanofluids have many applications in industry such as coolants, lubricants, heat exchangers, micro channel heat sinks and many others. The innovative technique, which uses a suspension of nanoparticles in the base fluid, was first introduced by Choi (1995) in order to develop advanced heat transfer fluids with substantially higher conductivities. There have been many studies in the literature to better understand the mechanism behind the enhanced heat transfer characteristics. An excellent collection of papers on this topic can be found in the books by Das et al. (2007) and in some review papers, for example, by Buongiorno (2006), Wang and Mujumdar (2007), and Kakaç and Pramuanjaroenkij (2009). Therefore, quite many investigators have studied the flow and thermal characteristics of nanofluids, both theoretically and experimentally.

For the flow over a shrinking sheet, the fluid is attracted towards a slot and as a result it shows quite different characteristics from the stretching case. From a physical point of view, vorticity generated at the shrinking sheet is not confined within a boundary layer and a steady flow is not possible unless either a stagnation flow or adequate suction is applied at the sheet. As discussed by Goldstein (1965), this new type of shrinking flow is essentially a backward flow. Miklavčič and Wang (2006) were the first who have investigated the flow over a shrinking sheet with suction effect. Steady two-dimensional and axisymmetric boundary layer stagnation point flow and heat transfer towards a shrinking sheet was analysed by Wang (2008). Thereafter various aspects of stagnation point flow and heat transfer over a shrinking sheet were investigated by many authors (Fang 2008; Fang et al. 2009; Ishak et al. 2010; Lok et al. 2011; Bhattacharyya 2011; Bhattacharyya et al. 2011; Nazar et al. 2011; Jafar et al. 2013). All studies mentioned above refer to the stagnation-point flow towards a stretching/shrinking sheet in a viscous and Newtonian fluid. Bachok et al. (2011) investigated the effects of solid volume fraction and the type of the nanoparticles on the fluid flow and heat transfer characteristics of a nanofluid over a stretching/shrinking sheet. On the other hand, Tham et al. (2012) considered the steady mixed convection boundary layer flow near the lower stagnation-point of a horizontal circular cylinder with a constant surface temperature embedded in a porous medium saturated by a nanofluid containing gyrotactic microorganisms.

In this paper, the steady stagnation-point flow towards a shrinking sheet in a nanofluid is theoretically studied. Three different types of nanoparticles, namely copper (Cu), alumina (Al2O3) and titania (TiO2) are considered. The nanofluid equations model as proposed by Tiwari and Das (2007) has been used. Although similar problem has been investigated, the main objective of the present paper is to extend the work of Nazar et al. (2011) by performing stability analysis of the dual solutions in order to determine which solution is stable and physically realisable.

2. Governing Equations

We consider the steady two-dimensional stagnation-point flow of a nanofluid past a shrinking

sheet with the linear velocity uw(x) = c x , and the velocity of the far (inviscid) flow is ( )eu x a x= , where a and c are constant, and x is the coordinate measured along the shrinking

surface. The flow takes place at 0y ≥ , where y is the coordinate measured normal to the shrinking surface. It is also assumed that the surface of the shrinking sheet is subjected to a

prescribed temperature ( )wT x T bx∞= + , where T∞ is the temperature of the ambient nanofluid, and b is a constant with 0b > (heated shrinking sheet). A steady uniform stress leading to equal and opposite forces is applied along the x − axis so that the sheet is stretched, keeping the origin fixed in a nanofluid.

Stability analysis of stagnation-point flow past a shrinking sheet in a nanofluid

53

The basic conservation of mass, momentum and energy equations for a nanofluid in Cartesian coordinates x and y are given by (Tiwari & Das 2007; Nazar et al. 2011)

∂u∂x

+ ∂v∂y

= 0

(1)

∂u∂t

+u ∂u∂x

+ v ∂u∂ y

= − 1ρnf

∂ p∂x

+µnfρnf

∂2u∂x2

+ ∂2u∂ y2

⎛

⎝⎜⎞

⎠⎟ (2)

∂v∂t

+u ∂v∂x

+ v ∂v∂ y

= − 1ρnf

∂ p∂ y

+µnfρnf

∂2v∂x2

+ ∂2v∂ y2

⎛

⎝⎜⎞

⎠⎟ (3)

∂T∂t

+u ∂T∂x

+ v ∂T∂ y

=α nf∂2T∂x2

+ ∂2T∂ y2

⎛

⎝⎜⎞

⎠⎟ (4)

subject to the initial and boundary conditions

t < 0 : u = 0, v = 0, T = T∞ for any x, yt ≥ 0 : v = 0, u = uw(x) = c x, T = Tw(x) at y = 0 u→ ue(x) = a x, T→T∞ as y→∞

(5)

Here u and v are the velocity components along the x and y axes, respectively, T is the temperature of the nanofluid, p is the fluid pressure, αnf is the thermal diffusivity of the

nanofluid, µnf is the effective viscosity of the nanofluid, and ρnf is the effective density of the nanofluid, which are given by (Oztop & Abu-Nada 2008)

α nf =knf

(ρCp )nf, µnf =

µ f(1−φ)2.5

, ρnf = (1−φ)ρ f +φ ρs ,

(ρCp )nf = (1−φ)(ρCp ) f +φ (ρCp )s ,knfk f

=(ks + 2k f ) − 2φ (k f − ks)(ks + 2k f ) +φ (k f − ks)

(6)

where φ is the solid volume fraction of the nanofluid or the nanoparticle volume fraction, ρ f

is the reference density of the fluid fraction, ρs is the reference density of the solid fraction, µ f

is the viscosity of the fluid fraction, knf is the effective thermal conductivity of the nanofluid,

k f is the thermal conductivity of the fluid, ks is the thermal conductivity of the solid, and

(ρCp )nf is the heat capacity of the nanofluid.


54

3. Steady-State Flow Case

We introduce now the following similarity variables:

η = (a /ν f )1/2 y, ψ = (aν f )

1/2 x f (η),θ(η) = (T −T∞ ) / (Tw −T∞ ),

(7)

where ν f is the kinematic viscosity of the fluid and the stream function ψ is defined in the

usual way as u = ∂ψ / ∂ y and v = −∂ψ / ∂x , which identically satisfy Eq. (1). On substituting (6) and (7) into Eqs. (2)-(4), we obtain the following nonlinear ordinary differential equations:

ε1 ′′′f + f ′′f +1− ′f 2 = 0 (8)

1Pr

ε2 ′′θ + f ′θ − ′f θ = 0

(9)

and the boundary conditions (5) become

f (0) = 0, ′f (0) = λ, θ(0) =1′f (∞) =1, θ(∞) = 0

(10)

where primes denote differentiation with respect to η , Pr =ν f /α f is the Prandtl number and λ = c / a is the stretching/shrinking parameter with λ > 0 for stretching and λ < 0 for

shrinking. Meanwhile ε1 and ε2 are two constants relating to the properties of the nanofluid, defined by

ε1 =1

(1−φ)2.5 1−φ +φ ρs / ρf( ) ,

ε2 =knf / kf

(1−φ) +φ (ρCp)s / (ρCp)f

(11)

The pressure p can be determined from Eq. (3) as

p = p0 − ρnf a2 x2 / 2 − ρnf v

2 / 2 + µnf ∂v / ∂ y (12)

where p0 0p is the stagnation pressure. It is worth mentioning that Eqs. (8) and (9) reduce to those first derived by Wang (2008) when the nanoparticle volume fraction parameter φ = 0 (regular Newtonian fluid), and Pr = 1.

The quantities of practical interest in this study are the skin friction coefficient at the

surface of the shrinking sheet fC and the local Nusselt number xNu , which can easily shown to be given by

Rex1/2Cf =

1(1−φ)2.5 ′′f (0), Rex

−1/2 Nux = −knfk f

′θ (0)

(13)

where Rex = ue(x)x /ν f is the local Reynolds number.


55

4. Stability Analysis

It has been shown in some papers (Weidman et al. 2006; Roşca & Pop 2013) that dual (lower and upper branch) solutions exist. In order to determine which of these solutions are physically realisable in practice, a stability analysis of Eqs. (8)-(10) is necessary. Following Weidman et al. (2006), a dimensionless time variable τ has to be introduced. The use of τ is associated with an initial value problem and this is consistent with the question of which solution will be obtained in practice (physically realisable). Thus, the new variables for the unsteady problem are

ψ = (aν f )1/2 x f (η,τ ), θ(η,τ ) = (T −T∞ ) / (Tw −T∞ ),

η = (a /ν f )1/2 y, τ = at

(14)

Substituting (14) into (2)-(4), we obtain

ε1∂3 f∂η3

+ f ∂2 f∂η2

+1− ∂ f∂η

⎛⎝⎜

⎞⎠⎟

2

− ∂2 f∂η∂τ

= 0

(15)

ε2Pr

∂2θ∂η2

+ f ∂θ∂η

− ∂ f∂η

θ − ∂θ∂τ

= 0

(16)

subject to the boundary conditions

f (0,τ ) = 0, ∂ f∂η(0,τ ) = λ, θ(0,τ ) =1

∂ f∂η(η,τ )→1, θ(η,τ )→ 0 as η→∞

(17)

To determine the stability of the solution f = f0(η) and θ = θ0(η) satisfying the boundary-value problem (8)-(10), we write (Weidman et al. 2006; Roşca & Pop 2013)

f (η,τ ) = f0(η) + e−γτ F(η,τ ),

θ(η,τ ) =θ0(η) + e−γ τ S(η,τ ),

(18)

where γ is an unknown eigenvalue parameter, while F(η,τ ) and S(η,τ ) are small relative

to f0(η) and θ0(η). Substituting (18) into (15) and (16), we obtain the following linearised problem:

ε1∂3F∂η3

+ f0∂2F∂η2

+ Ff0 '' − 2 f0 '∂F∂η

+γ ∂F∂η

− ∂2F∂η∂τ

= 0

(19)

ε2Pr

∂2 S∂η2

+ f0∂S∂η

+ Fθ0 ' − f0 'S −∂F∂η

θ0 +γ S −∂S∂τ

= 0

(20)

subject to the boundary conditions


56

F(0,τ ) = 0, ∂F∂η(0,τ ) = 0, S(0,τ ) = 0

∂F∂η(η,τ )→ 0, S(η,τ )→ 0 as η→∞

(21)

As suggested by Weidman et al. (2006), we investigate the stability of the steady flow and

heat transfer solution f0(η) and θ0(η) by setting τ = 0 , and hence F = F0(η) and S = S0(η) in (19) and (20) to identify initial growth or decay of the solution (18). To test our numerical procedure we have to solve the following linear eigenvalue problem:

ε1F0 ''' + f0 F0 ''+ F0 f0 '' − 2 f0 'F0 '+γ F0 ' = 0 (22)

ε2PrS0 '' + f0 S0 ' + F0θ0 ' − f0 'S0 − F0 'θ0 +γ S0 = 0

(23)

with the boundary conditions

F0(0) = 0, F0 '(0) = 0, S0(0) = 0F0 '(η)→ 0, S0(η)→ 0 as η→∞

(24)

It should be mentioned that for particular values of λ,θ , and other parameters involved, the

corresponding steady flow solution f0(η) and θ0(η) , the stability of the steady flow solution

is determined by the smallest eigenvalue γ 1 . According to Harris et al. (2009), the range of

possible eigenvalues can be determined by relaxing a boundary condition on F0 '(η) or S0(η) .

For the present problem, we relax the condition that F '0 (η)→ 0 as η→∞ and for a fixed

value of γ we solve the system (22-23) along with the new boundary condition F0 ''(0) = 1.

5. Results and Discussion

The system of nonlinear ordinary differential equations (8) and (9) subject to the boundary conditions (10) are solved numerically using the bvp4c function in Matlab. Following Oztop and Abu-Nada (2008), we considered the range of nanoparticle volume fraction to be 0 ≤φ ≤ 0.2 . The Prandtl numbers Pr considered in this study are Pr = 1 (for comparison) and Pr = 6.2 (water-based nanofluid). Further, it should also be pointed out that the thermophysical properties of fluid and nanoparticles (Cu, Al2O3, TiO2) used in this study are given in Table 1 (Oztop & Abu-Nada 2008).

On the other hand, it is worth mentioning that the present study reduces to regular (Newtonian) fluid when φ = 0 . The dual (first and second) solutions are obtained by setting different initial guesses for the missing values of f ''(0) and θ '(0) , where all profiles satisfy the far field boundary conditions (10) asymptotically. In order to validate the present numerical method used, we have compared our results with those obtained by Nazar et al. (2011) for various values of the shrinking parameter (λ < 0 ) when φ = 0.2 and Pr = 1 as shown in

Tables 2 and 3. Table 2 presents the values of the skin friction coefficient Rex1/2Cf while


57

Table 3 illustrates the values of the local Nusselt number Rex−1/2 Nux for shrinking sheets. The

comparisons with Nazar et al. (2011) are found to be in excellent agreement. Therefore, the present numerical solution is further validated, so that we are confident our results are accurate. Further, Tables 2 and 3 also present the numerical values for the present problem in nanofluid with different nanoparticles (Cu, Al2O3, TiO2) for various nanoparticle volume fraction φ.

Table 1: Thermophysical properties of fluids and nanoparticles (Oztop & Abu-Nada 2008)

Physical properties

Fluid phase (water)

Cu Al2O3 TiO2

Cp (J/kg K) 4179 385 765 686.2

r (kg/m3) 997.1 8933 3970 4250k (W/mK) 0.613 400 40 8.9538

Table 2: Value of Rex1/2Cf for shrinking sheet (λ < 0) with φ = 0.2 (results in [ ] are those of Nazar et al. (2011))

lUpper branch Lower branch

Cu Al2O3 TiO2 Cu Al2O3 TiO2

-0.25 2.98374 [2.98374]

2.34163 [2.34163]

2.38247 [2.38247]

-[ - ]

-[ - ]

-[ - ]

-0.50 3.18254[3.18254]

2.49765[2.49765]

2.54121[2.54121]

-[ - ]

-[ - ]

-[ - ]

-0.75 3.16898[3.16898]

2.48701[2.48701]

2.53038[2.53038]

-[ - ]

-[ - ]

-[ - ]

-1.00 2.82750[2.82750]

2.21902[2.21902]

2.25772[2.25772]

-[ - ]

-[ - ]

-[ - ]

-1.10 2.52506[2.52506]

1.98166[1.98166]

2.01622[2.01622]

0.10475[0.10475]

0.08221[0.08221]

0.08364[0.08364]

-1.15 2.30281[2.30281]

1.80724[1.80724]

1.83876[1.83876]

0.24832[0.24832]

0.19488[0.19488]

0.19828[0.19828]

-1.20 1.98415[1.98415]

1.55716[1.55716]

1.58432[1.58431]

0.49717[0.49717]

0.39018[0.39018]

0.39698[0.39698]

Table 3: Value of Rex−1/2 Nux for shrinking sheet (λ < 0) with φ = 0.2 and Pr=1

(results in [ ] are those of Nazar et al. (2011))

lUpper branch Lower branch


-0.25 1.03949 [1.03949]

0.93681 [0.93681]

0.89501 [0.89501]

-[ - ]

-[ - ]

-[ - ]

-0.50 0.87555 [0.87555]

0.75219 [0.75219]

0.71593 [0.71593]

-[ - ]

-[ - ]

-[ - ]

-0.75 0.67521 [0.67521]

0.52533 [0.52533]

0.49578[0.49578]

-[ - ]

-[ - ]

-[ - ]

-1.00 0.40000 [0.40000]

0.21058 [0.21058]

0.19002 [0.19002]

-[ - ]

-[ - ]

-[ - ]

-1.10 0.23996 [0.23996]

0.02522 [0.02522]

0.00975 [0.00975]

-2.28730[-2.27462]

-3.46470[-3.46468]

-3.42839[-3.42772]

to be continued...


58

-1.15 0.13343 [0.13343]

-0.09941[-0.09941]

-0.11156[-0.11156]

-1.51080[-1.51080]

-2.27652[-2.27652]

-2.25087[-2.25086]

-1.20 -0.01501[-0.01501]

-0.27512[-0.27512]

-0.28277[-0.28277]

-1.01271[-1.01270]

-1.55706[-1.55706]

-1.53989[-1.15162]

It is shown in Tables 2 and 3 the variations of the skin friction coefficient Rex1/2Cf and

the local Nusselt number Rex−1/2 Nux , respectively, with λ < 0 (shrinking sheet) for different

nanoparticles (Cu, Al2O3, TiO2) when φ =0.2 . As can be seen clearly in Figures 1 to 4, it is

found that dual solutions exist when λ < −1, up to a certain critical value of l, say λc , beyond which the boundary layer separates from the surface, thus no solution is obtained. In this study,

the value of λc is found to be λc = 1.2465, which is consistent with the value obtained in Nazar et al. (2011) and Wang (2008) for φ = 0 (regular Newtonian fluid). We identify the upper (first) and lower (second) branch solutions in the following discussion by how they appear in Figures 1 to 4, i.e. the upper branch solution has higher values for a given l than the lower branch solution. It is also shown in Figures 1 and 2 that for a particular nanoparticle used (Cu as in example), as the nanoparticle volume fraction φ increases, the skin friction coefficient and the local Nusselt number also increase.

-1.5 -1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

l

Re x1/

2 Cf

Upper branchLower branch

φ=0, 0.1, 0.2

λc= - 1.2465

Figure 1: Variation of the skin friction coefficient 1/2xRe fC with l for different value of φ in Cu-water nanofluid

......continuation


59

-1.5 -1 -0.5 0 0.5 1-30

-25

-20

-15

-10

-5

0

5

l

Re x-1

/2N

u


φ = 0, 0.1, 0.2

λc= - 1.2465

Figure 2: Variation of the local Nusselt number 1/ 2xRe xNu− with l for different value of φ in Cu-water nanofluid

when Pr = 6.2

-1.5 -1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

l

Re x1/

2 Cf


Cu, TiO2, Al2O3λ

c= - 1.2465

Figure 3: Variation of the skin friction coefficient 1/2xRe fC with l for different types of nanoparticle

when φ = 0.2


60

-1.5 -1 -0.5 0 0.5 1-30

-25

-20

-15

-10

-5

0

5

l

Re x-1

/2N

u


λc= - 1.2465

Cu, Al2O3, TiO2

Figure 4: Variation of the local Nusselt number 1/ 2xRe xNu− with l for different types of nanoparticle when

φ = 0.2 and Pr = 6.2

On the other hand, Figures 3 and 4 show that for a fixed value of φ , say φ = 0.2 , the skin friction is highest for nanoparticle Cu (nanoparticle with high density), followed by TiO2 and Al2O3, while the local Nusselt number is largest for Cu, followed by Al2O3 and TiO2 (nanoparticle with low thermal conductivity). Figure 5 displays the velocity profiles ′f (η) for nanoparticles Cu, Al2O3 and TiO2 when φ = 0.2 and l = -1.2 (shrinking sheet), while Figure 6 illustrates the corresponding temperature profiles when Pr = 6.2. The velocity and temperature profiles for Al2O3 and TiO2 are almost identical while the velocity for nanoparticle Cu is the highest but the temperature for Cu is the lowest. This is consistent with the variation of the reduced skin friction coefficient (0)f ′′ which is highest for Cu and the variation of the temperature gradient or heat transfer rate − ′θ (0)which is highest for TiO2. Therefore, it is found that nanoparticles with low thermal conductivity, TiO2, have better enhancement on heat transfer compared to Al2O3 and Cu. These velocity and temperature profiles in Figures 5 and 6 support the existence of dual nature of the solutions presented in Figures 1 to 4. It is seen that the lower branch solutions have larger boundary layer thickness compared to the upper branch solutions. These profiles satisfy the far field boundary conditions (10) asymptotically, which support the numerical results obtained and thus, cannot be neglected mathematically.


61

0 1 2 3 4 5 6 7 8

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

h

f '(h

)


Al2O3, TiO2, Cu

Figure 5: Velocity profiles ′f (η) for different types of nanoparticle when φ = 0.2 and λ = -1.2 (shrinking sheet)

0 1 2 3 4 5 6 7 80

2

4

6

8

10

12

14

h

q (h

)

Upper branchLower branch TiO2, Al2O3, Cu

Figure 6: Temperature profiles θ (η) for different types of nanoparticle when f= 0.2, λ = -1.2 (shrinking sheet) and Pr = 6.2

We can expect that the upper branch solution is stable, while the lower branch solution is not, since the upper branch is the only solution for the case λ ≥ −1. In order to validate this, a stability analysis is carried out by solving linear eigenvalue problem (22-23) subject

to homogeneous boundary conditions (24). The smallest eigenvalue γ 1 for some values of φ and λ are shown in Table 4 when Pr = 6.2 for different type of nanoparticles. The results

show that all the upper branch solutions have positive eigenvalueγ 1 while all the lower

branch solutions have negative eigenvalue γ 1 . We conclude that the upper branch solution is physically realisable (stable) while the lower branch is not physically realisable (unstable) over

time. Notice that the smallest eigenvalue γ 1 is unique when different types of nanoparticles or different nanoparticle volume fraction parameters φ are used since these parameters do not

affect the critical value λc as shown in Figures 1 to 4, due to the uncoupled Eqs. (8) and (9). We can also expect the same observation when different Prandtl number Pr is used.


62

Table 4: Value of smallest eigenvalue γ 1 for some value of φ and λ

φ lUpper branch Lower branch


0.1 -1.05 1.2180 1.2180 1.2180 -0.9363 -0.9363 -0.9363-1.10 1.0463 1.0463 1.0463 -0.8437 -0.8437 -0.8437-1.15 0.8429 0.8429 0.8429 -0.7135 -0.7135 -0.7135-1.20 0.5780 0.5780 0.5780 -0.5172 -0.5172 -0.5172-1.24 0.2121 0.2121 0.2121 -0.2036 -0.2036 -0.2036

0.2 -1.05 1.2180 1.2180 1.2180 -0.9363 -0.9363 -0.9363-1.10 1.0463 1.0463 1.0463 -0.8437 -0.8437 -0.8437-1.15 0.8429 0.8429 0.8429 -0.7135 -0.7135 -0.7135-1.20 0.5780 0.5780 0.5780 -0.5172 -0.5172 -0.5172-1.24 0.2121 0.2121 0.2121 -0.2036 -0.2036 -0.2036

6. Conclusions

We have studied numerically the problem of steady two-dimensional laminar stagnation-point flow towards a shrinking sheet in nanofluids. The governing partial differential equations are transformed into nonlinear ordinary differential equations via the similarity transformation, before being solved numerically using bvp4c function in Matlab. The comparison of the results with previously published results shows an excellent agreement thus gives us confidence in our computations presented in this paper. Numerical results were obtained for the skin friction coefficient, the local Nusselt number as well as the velocity and temperature profiles for some values of the governing parameters, namely the nanoparticle volume fraction φ (0 ≤φ ≤ 0.2 ), the shrinking parameter l and the Prandtl number Pr. It was found that nanoparticles with low thermal conductivity (TiO2) have better enhancement on heat transfer compared to those with higher thermal conductivity (Al2O3 and Cu). For a particular nanoparticle, skin friction coefficient and the heat transfer rate at the surface increased as the nanoparticle volume fraction φ increased. It was also found that solutions do not exist for larger shrinking rates and dual solutions exist when λ < −1.0 for both a regular fluid (φ = 0 ) and for a nanofluid (φ ≠ 0 ). The stability analysis was performed and showed that the upper branch solution is stable while the lower branch solution is unstable.

Acknowledgement

This work was supported by the research grant (AP-2013-009) from the Universiti Kebangsaan Malaysia.

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1School of Mathematical SciencesFaculty of Science & TechnologyUniversiti Kebangsaan Malaysia43600 UKM BangiSelangor DE, MALAYSIAE-mail: [email protected], [email protected]*

2Faculty of Engineering & Built Environment Universiti Kebangsaan Malaysia43600 UKM BangiSelangor DE, MALAYSIAE-mail: [email protected]

* Corresponding author

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